Properties

Label 619.12
Modulus $619$
Conductor $619$
Order $618$
Real no
Primitive yes
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(619, base_ring=CyclotomicField(618))
 
M = H._module
 
chi = DirichletCharacter(H, M([77]))
 
pari: [g,chi] = znchar(Mod(12,619))
 

Basic properties

Modulus: \(619\)
Conductor: \(619\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(618\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 619.h

\(\chi_{619}(2,\cdot)\) \(\chi_{619}(10,\cdot)\) \(\chi_{619}(11,\cdot)\) \(\chi_{619}(12,\cdot)\) \(\chi_{619}(15,\cdot)\) \(\chi_{619}(18,\cdot)\) \(\chi_{619}(19,\cdot)\) \(\chi_{619}(21,\cdot)\) \(\chi_{619}(32,\cdot)\) \(\chi_{619}(40,\cdot)\) \(\chi_{619}(46,\cdot)\) \(\chi_{619}(48,\cdot)\) \(\chi_{619}(52,\cdot)\) \(\chi_{619}(55,\cdot)\) \(\chi_{619}(56,\cdot)\) \(\chi_{619}(59,\cdot)\) \(\chi_{619}(62,\cdot)\) \(\chi_{619}(65,\cdot)\) \(\chi_{619}(66,\cdot)\) \(\chi_{619}(67,\cdot)\) \(\chi_{619}(68,\cdot)\) \(\chi_{619}(69,\cdot)\) \(\chi_{619}(70,\cdot)\) \(\chi_{619}(75,\cdot)\) \(\chi_{619}(76,\cdot)\) \(\chi_{619}(78,\cdot)\) \(\chi_{619}(82,\cdot)\) \(\chi_{619}(83,\cdot)\) \(\chi_{619}(84,\cdot)\) \(\chi_{619}(85,\cdot)\) ...

sage: chi.galois_orbit()
 
order = charorder(g,chi)
 
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Related number fields

Field of values: $\Q(\zeta_{309})$
Fixed field: Number field defined by a degree 618 polynomial (not computed)

Values on generators

\(2\) → \(e\left(\frac{77}{618}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 619 }(12, a) \) \(-1\)\(1\)\(e\left(\frac{77}{618}\right)\)\(e\left(\frac{71}{206}\right)\)\(e\left(\frac{77}{309}\right)\)\(e\left(\frac{268}{309}\right)\)\(e\left(\frac{145}{309}\right)\)\(e\left(\frac{122}{309}\right)\)\(e\left(\frac{77}{206}\right)\)\(e\left(\frac{71}{103}\right)\)\(e\left(\frac{613}{618}\right)\)\(e\left(\frac{47}{618}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 619 }(12,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 619 }(12,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 619 }(12,·),\chi_{ 619 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 619 }(12,·)) \;\) at \(\; a,b = \) e.g. 1,2